Analytic surgery and analytic torsion

Abstract

Analytic surgery, as defined in [9] and [6], is a one-parameter metric deformation of a Riemannian manifold M , which stretches M across a separating hypersurface H in a cylindrical fashion; the singular limit is a complete manifold with asymptotically cylindrical ends, M . In this paper, the analysis of [9] and [6] is used to study the behaviour of analytic torsion of unitary representations under analytic surgery. A gluing formula is obtained relating the analytic torsion of M to the ‘b-analytic torsion’ bT (a regularized analytic torsion on manifolds with boundary) of M . This is then used to prove the Cheeger-Muller theorem, asserting the equality of analytic and Reidemeister torsion τ on closed manifolds, and to prove the following combinatorial formula for b-analytic torsion on odd dimensional manifolds with boundary: b T (N) = τ(N)2 . As a step in the proof, a Hodge-theoretic description of the Mayer-Vietoris sequence for cohomology under analytic surgery is developed.